An approach to the errors-in-variables regression model

This paper concerns with estimation of the errors-in-variable regression model where the explanatory variables contain measurement errors. It is well known that the ordinary least squares (OLS) estimator of the errors-in-variables model is asymptotically downward biased (inconsistent). There have been various methods proposed in order to handle the problem. They essentially assume some a priori information; (1) the ratio(s) of variances of latent explanatory variable(s) and measurement error(s),  or (2)  availability of suitable instrumental variables.

In this paper, we alternatively assume that latent explanatory variables are positively autocorrelated. Since most economic variables are positively  autocorrelated, this assumption is not restrictive.

  The first purpose of this paper is to analytically show that the temporal aggregation (i.e., aggregation over time) of the model gives an alternative scheme to circumvent the problem caused by errors-in-variables. The basic idea comes from the observation that temporal aggregation of positively autocorrelated variables increases their variability faster than non-autocorrelated measurement errors, and it decreases the bias and mean squared error (MSE) of the OLS estimator.   

The second purpose of this paper is to propose a convenient consistent estimator. The temporal aggregation gives an alternative set of estimates in addition to the estimates of the (original) disaggregated model, and two sets of estimates can be combined to derive a consistent estimator of the parameters of the errors-in-variables model. It is expected that the proposed estimator performs better than the conventional consistent estimators, since the former effectively contains more information.

Temporal aggregation has been widely discussed in the context of forecasting performance of pure time-series models rather than regression models. They generally show unfavorable results as compared with those of temporally disaggregated models. The result of the present paper is rather unique in producing favorable results based upon temporal aggregation.  

The suitably designed Monte Carlo experiment shows that the analytical results are supported even in small samples and the proposed estimator is superior to the conventional consistent estimators.

Finally, the method proposed in the paper is applied to estimate the Fisher equation in Japanese economy. The result seems to suggest that the proposed estimator substantially (although not completely) eliminates the downward bias caused by errors-in-variables.